I have asked this question on math stack exchange yesterday, but there is no reply. 
See https://math.stackexchange.com/questions/4818719/does-there-exist-research-about-equation-like-u-tt-detd2u

Does there exist research about equation like $$u_{tt}=\det(D_{x}^{2}u)+\dots$$? That is to say, it contains second order time term $u_{tt}$ and the determination of Hessian of solution $\det(D_{x}^{2}u)$. Recently, I have read some papers concerning Hyperbolic meaning curvature flow like the paper**The hyperbolic mean curvature flow by K.Smoczyk and philippe G.Lefloch on JMPA**. But does there exist some research like the above question, if yes, can you offer me some paper? Thank advance.